Chapter 5: Q. 5.20 (page 216)

Verify that $Var (X) = \frac{α}{λ^{2}}$ when $X$ is a gamma random variable with parameters $α$ and $λ$

Short Answer

Expert verified

To get the Variance, find the generic expression for $E (X^{k})$ and use it.

Step by step solution

Determine the positive integral.

Let's find the first and second moments of $X$ if $X$ $~$ Gamma $(α, λ)$

For any positive integer $k$ .

we have that

localid="1649619459546" $E (X^{k}) = \int_{0}^{\infty} x^{k} \cdot \frac{λ^{α}}{Γ (α)} x^{α - 1} e^{- λ x} d x = \frac{λ^{α}}{Γ (α)} \int_{0}^{\infty} x^{k + α - 1} e^{- λ x} d x$

Expression the integral.

in order to evaluate the integral, make the substitution $s = λ x$

Hence, the expression above is equal to localid="1649619493912" $\frac{λ^{α}}{Γ (α)} \int_{0}^{\infty} x^{k + α - 1} e^{- λ x} d x = \frac{λ^{α}}{Γ (α)} \int_{0}^{\infty} {(\frac{s}{λ})}^{k + α - 1} e^{- s} \frac{d s}{λ}$

localid="1649619476293"

= \frac{1}{λ^{k} \cdot Γ (α)} \int_{0}^{\infty} s^{k + α - 1} e^{- s} d s = \frac{Γ (k + α)}{λ^{k} \cdot Γ (α)}

So, we have obtained that,

$E (X^{k}) = \frac{Γ (k + α)}{λ^{k} \cdot Γ (α)}$

Implies the value.

The implies that

$E (X) = \frac{Γ (α + 1)}{λ \times Γ (α)} = \frac{α \times Γ (α)}{λ \times Γ (α)} = \frac{α}{λ}$

and

E (X^{2}) = \frac{Γ (α + 2)}{λ^{2} \times Γ (α)} = \frac{(α + 1) α Γ (α)}{λ^{2} \times Γ (α)} = \frac{(α + 1) α}{λ^{2}}

Finally, we have that

$Var (X) = E (X^{2}) - E (X)^{2} = \frac{(α + 1) α}{λ^{2}} - \frac{α^{2}}{λ^{2}} = \frac{α}{λ^{2}}$

Which has to be proved.

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Verify that $Var (X) = \frac{α}{λ^{2}}$ when $X$ is a gamma random variable with parameters $α$ and $λ$

Short Answer

Step by step solution

Determine the positive integral.

Expression the integral.

Implies the value.

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Verify thatVar(X)=αλ2whenXis a gamma random variable with parameters αand λ

Short Answer

Step by step solution

Determine the positive integral.

Expression the integral.

Implies the value.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Theoretical and Mathematical Physics

Statistics

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.

Verify that $Var (X) = \frac{α}{λ^{2}}$ when $X$ is a gamma random variable with parameters $α$ and $λ$